Un hook the motor and Connect a known load like a light bulb across the wires from the switch. Flip switch in both directions. Does lamp light in both positions? Is it the same brightness?
I didn't see this suggestion (next page) until after I tried to follow Rivets' suggestion. When I took the connector off the switch to check continuity, I accidentally took the switch apart. I put it back together, tried to check continuity again, but realized the reason I wasn't getting anything was because I'd disconnected the battery, so some downstream relay probably wasn't operating. Put the whole thing back together, tried it out, and everything works, both new and old motors. I'm guessing that the issue was some connection either internal to the switch or the connector to it, was making contact but not enough to drive the motor. The suggested bulb test probably would have shown me that.
Now I'll find out if Flip Manufacturing really takes returns at no cost for any reason as their website says. Otherwise, I guess I pay the stupid tax!